Video: AQA GCSE Mathematics Higher Tier Pack 5 β€’ Paper 1 β€’ Question 18

This prism has a cross section consisting of a semicircle attached to a parallelogram. All the lengths in the diagram are in centimetres. Find an expression for the total volume of the prism in cubic centimetres. Write your answer in the form π‘Ž(π‘₯Β³ + π‘₯Β²) + π‘πœ‹π‘₯Β³.

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Video Transcript

This prism has a cross section consisting of a semicircle attached to a parallelogram. All the lengths in the diagram are in centimetres. Find an expression for the total volume of the prism in cubic centimetres. Write your answer in the form π‘Ž multiplied by π‘₯ cubed plus π‘₯ squared plus 𝑏 multiplied by πœ‹ multiplied by π‘₯ cubed.

Let’s begin by reminding ourselves what the formula is for the volume of a prism. It’s the area of its cross section multiplied by its length. The cross section of this prism is a compound shape. It’s made up of a semicircle and a parallelogram. We do know its length though. It’s one-half π‘₯.

Let’s begin by working out the area of the cross section. We’re going to calculate the area of the two shapes it’s made up of. The formula for area of a circle is πœ‹π‘Ÿ squared, where π‘Ÿ is the radius. So we can find the area of a semicircle by halving this. We can find the diameter of our semicircle. Since opposite sides of a parallelogram are equal in length, this means the diameter of our semicircle is π‘₯. The radius is half of the length of the diameter. So it’s a half of π‘₯ or π‘₯ over two. The area of the semicircle is therefore given by πœ‹ multiplied by the radius squared. That’s π‘₯ over two squared all over two.

We need to be careful when squaring π‘₯ over two. We have to square both the numerator and the denominator. That gives us π‘₯ squared over four. Having a fraction within a fraction made that last sum a little bit tricky. Instead, if we write it as a half of πœ‹ multiplied by π‘₯ squared over four, it’s much easier to deal with.

We multiply the numerators. That’s one multiplied by πœ‹ multiplied by π‘₯ squared. That’s πœ‹π‘₯ squared. We then multiply the denominators, remembering that the denominator of πœ‹ will be one. Two multiplied by one multiplied by four is eight. So the area of the semicircle is πœ‹ multiplied by π‘₯ squared over eight or an eighth of πœ‹π‘₯ squared.

Next, we need to find the area of the parallelogram. That’s found by multiplying its base by its perpendicular height. The base of our parallelogram is π‘₯ and its height is π‘₯ plus one. So its area is π‘₯ multiplied by π‘₯ plus one. It’s important to include the brackets around the π‘₯ plus one because that reminds us that the π‘₯ is multiplying both the π‘₯ on the inside of that bracket and by the one. π‘₯ multiplied by π‘₯ is π‘₯ squared and π‘₯ multiplied by one is π‘₯. So the area of the parallelogram is π‘₯ squared plus π‘₯.

We can see then that the total area of the cross section is an eighth of πœ‹ multiplied by π‘₯ squared plus another π‘₯ squared plus π‘₯. All that’s left is to multiply this area by the length of the prism. That’s one-half π‘₯. We’ll multiply this bracket out by multiplying each term inside the bracket by one-half π‘₯. One-eighth multiplied by one-half is one sixteenth. So we get one sixteenth of πœ‹π‘₯ cubed. π‘₯ squared multiplied by a half π‘₯ is a half π‘₯ cubed and π‘₯ multiplied by a half π‘₯ is a half π‘₯ squared.

We do need to write our answer in the form given. Notice how π‘₯ cubed and π‘₯ squared both have a coefficient of a one-half. We can therefore factorise this partially by taking out a factor of one-half. And a half π‘₯ cubed plus a half π‘₯ squared is the same as a half multiplied by π‘₯ cubed plus π‘₯ squared. And we’re left also with one sixteenth of πœ‹π‘₯ cubed. Comparing this to our form given in the answer, we can see that π‘Ž is equivalent to a half and 𝑏 is equivalent to one sixteenth.

And we’re done! The volume of our prism is a half multiplied by π‘₯ cubed plus π‘₯ squared plus a sixteenth πœ‹π‘₯ cubed.

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